Remark 3 The FunPrinSS probability (REF ) is closely related to the leverage sampling probability [1]} for classical linear regression. Let \(\mathbf {X} = \left(\mathbf {x}_1,\cdots ,\mathbf {x}_N\right)^T\)  be a \(N\times p\)  fixed design matrix with \(N\)  observations and \(p\)  variables (with \(N>p\) ).
Suppose the singular value decomposition (SVD) of \(\mathbf {X}\)  is \(\mathbf {X}= \mathbf {U}\mathbf {D}\mathbf {V}^T\) , where \(\mathbf {D}\)  is a diagonal matrix with singular values, and the columns of \(\mathbf {U}\)  and \(\mathbf {V}\)  contain the left and right singular vectors, respectively. Denote \(U_{nj}\)  as the \((n,j)\) -th element of \(\mathbf {U}\) .
The leverage sampling probability \(p_n\)  is proportional to the statistical leverage \(h_{nn}\)  of \(n\) -th observation, i.e.
\(p_n\propto h_{nn} = \mathbf {x}_n^T\left(\mathbf {X}^T\mathbf {X}\right)^{-}\mathbf {x}_n= \sum _{j = 1}^p U_{nj}^2.\)  When the samples \(\mathbf {x}_n\) 's have zero mean, \(U_{nj}\)  can be interpreted as multivariate principal component scores, and the statistical leverage \(h_{nn}\)  is the summation of all squared scores in the \(p\) -dimensional space. In contrast, our FunPrinSS probability (REF ) makes a rank-\(R\)  truncation to the Karhunen-Loève expansion (REF ) of the infinite dimensional functional data, and takes into account the effect of the remaining scores via the projected residual \((I - \widehat{\mathcal {P} }_R) x_n\) .

